Impartida por: Mauricio Misquero Castro (Universidad de Granada – FPU)
Resumen: It is studied a Hill’s equation, depending on two parameters $e\in [0,1)$ and $\Lambda>0$, that has applications to some problems in Celestial Mechanics of the Sitnikov-type. Due to the nonlinearity of the eccentricity parameter $e$ and the coexistence problem, the stability diagram in the $(e,\Lambda)$-plane presents unusual resonance tongues emerging from points $(0,(\frac{n}{2})^2),\ n=1,2,…$. The tongues bounded by curves of eigenvalues corresponding to $2\pi$-periodic solutions collapse into a single curve of coexistence (for which there exist two independent $2\pi$-periodic eigenfunctions), whereas the remaining tongues have no pockets and are very thin.

Unlike most of the literature related to resonance tongues and Sitnikov-type problems, the study of the tongues is made from a global point of view in the whole range of $e\in[0,1)$. We apply the stability diagram of our equation to determine the regions for which the equilibrium of a Sitnikov $(N+1)$-body problem is stable in the sense of Lyapunov and the regions having symmetric periodic solutions with a given number of zeros.

Resumen: I will introduce Lyapounov functions for cone fields, a generalization of the causal structure of a Lorentzian metric, and present some results on their existence. If time permits I will define a notion of global hyperbolicity for cone fields and give a result on the existence of steep Lyapounov functions for globally hyperbolic cone fields. The material is a cooperation with Patrick Bernard (Université Paris Dauphine).

Título: Geodesic completeness of compact Lorentzian manifoldsResumen: A semi-Riemannian manifold is geodesically complete (or for short, complete) if its maximal geodesics are defined for all times. For Riemannian metrics the compactness of the manifold implies completeness. In contrast, there Lorentzian metrics on the torus that are not complete. Nevertheless, completeness plays an important role for fundamental geometric questions in Lorentzian geometry such as the classification of compact Lorentzian symmetric spaces and in particular for a Lorentzian version of Bieberbach’s theorem. We will study the completeness for compact manifolds that arise from the classification of Lorentzian holonomy groups, which we will briefly review in the talk. These manifolds have abelian holonomy and carry a parallel null vector field. By determining their universal cover we show that they are complete. In the talk we will explain this result and further work in progress, both being joint work with A. Schliebner (Humboldt-Universität zu Berlin).